Let g(x) = -2x2 + bx + c be a quadratic function, defined everywhere, where b and c are constants. If x = 1 marks the location of one of the zeros of this quadratic function, and if the y-intercept of this function is at (0, 5), then use this information to name constant b.
g = -2x^2 + bx + c
g = (x-1)(-2x + b-2) remainder b+c-2
In order for 1 to be a root, then x-1 must divide g exactly. That is,
b+c-2 = 0
Using the y-intercept, when x=0, y=5
5 = c
So, b+3 = 0, or b=-3
g = -2x^2 -3x + 5 = (x-1)(-2x - 5)
To find the constant b, we can use the fact that x = 1 marks the location of one of the zeros of the quadratic function. This means that if we substitute x = 1 into the function g(x), it should result in g(1) = 0.
So, let's substitute x = 1 into the equation:
g(x) = -2x² + bx + c
g(1) = -2(1)² + b(1) + c
= -2 + b + c
Since we know that g(1) = 0, we can set -2 + b + c = 0:
-2 + b + c = 0
Now, we also know that the y-intercept of this function is at (0, 5). The y-intercept is the point where x = 0, so let's substitute x = 0 and y = 5 in the equation:
g(x) = -2x² + bx + c
g(0) = -2(0)² + b(0) + c
= 0 + 0 + c
= c
Since the y-intercept is at (0, 5), we have c = 5.
Substituting c = 5 into the previous equation -2 + b + c = 0:
-2 + b + 5 = 0
Simplifying the equation:
b + 3 = 0
Subtracting 3 from both sides:
b = -3
Therefore, the constant b is -3.